Lesson 6 | The Basic Reproduction Number R0
Introduction
The basic reproduction number (R0) is an indicator of the transmissibility of infectious agents. It represents the number of new infections estimated to stem from a single case in a population that is fully susceptible to infection.
It is used to answer the question: under what conditions will an infectious disease invade a system?
Calculating R0 can inform public health policy decisions to prevent and control epidemics.
Let’s get started!
Learning Objectives
By the end of this lesson, you should be able to:
Relate population biology to the study of epidemic dynamics.
Define the basic reproduction number R0 for infectious diseases.
Learn the history and importance of R0 in epidemiology.
Understand and explain the significance of pathogen invasion thresholds.
Calculate R0 for a given infectious disease model.
Derive R0 from a system of SIR model ODEs.
Understand the implications, assumptions, and limitations of R0.
1 Basic Reproduction Number (R0)
1.1 Defining R0
The basic reproduction number (R0) is defined as the number of secondary cases generated by one infected case in an entirely susceptible population.
For example, if the R0 is 2, then one infected person will infect, on average, two new people before they recover.
The definition assumes that there is only one infectious individual at the start, and no other individuals are infected or immunized.
It determines whether a pathogen can invade and spread in a population.
- If \(R_0 > 1\), the infection is
expected to spread and “invade” the population, causing an epidemic.
- If \(R_0 < 1\), the infection is expected to die out and go “extinct”, because on average an infectious person is expected to transmit to fewer than 1 other susceptibles.
R0 Terminology and Notation
The basic reproduction number (R0) is also known as the basic reproduction ratio or the basic reproductive rate. It is generally pronounced “R nought,” but you may also hear it being pronounced as “R zero”.
The symbol R stands for reproduction. Naught, or zero, stands for the zeroth generation (patient zero). It refers to the first patient infected by a disease in an epidemic, also referred to as the index case.
Practice question 1
Answer TRUE or FALSE to the following statement:
When \(R_0 = 2\) then each infected person will go on to infect exactly 2 susceptible individuals.
2 History of the Basic Reproduction Number (R0)
The concept of a reproduction number (or ratio) was first introduced in the field of mathematical demography, which is the study of population dynamics. In demography and ecology, R0 is a measure of reproduction rate in a group of people or animals and used it to count offspring.
In the 1950s, epidemiologist George MacDonald suggested using R0 to describe the transmission potential of malaria. When R0 was adopted for use by epidemiologists, the objects being counted were infected cases. He proposed that if R0 is less than 1, the disease will die out in a population, because on average an infectious person will transmit to fewer than 1 other susceptible person. On the other hand, if R0 is greater than 1, the disease will spread (Eisenberg, 2020). Since then the reproduction number has become widely used in the field of epidemiology.
3 What is an epidemic?
Epidemic refers to an unexpected increase, often sudden, in the number of disease cases above what is normally expected in that population in that area.
Centers for Disease Control and Prevention (CDC)
Let’s take a closer look at the trajectories of each of these epidemic curves:
We can identify key stages of an epidemic:
Key questions are:
Why did an epidemic start? What caused the sudden change in incidence or prevalence?
When can an infectious disease establish? What was it about this particular scenario that made it possible for the pathogen to invade?
When can the disease persist? Once the pathogen emerges, will it be able to sustain the epidemic for a long time? Or will it cause a few cases and then fade out?
Can we eradicate the disease? How can we prevent a pathogen from invading a population?
3.1 Moving from Individual to Population Scale
Let’s revisit our classic SIR model framework:
When developing a compartmental model, we focus on interactions at the individual level.
We categorize individuals into compartments based on their infection or immune status: Susceptible (S), Infected (I), or Recovered (R). Using a system of ODEs, we incorporate parameters for transmission (β) and recovery (γ) to determine the rate of “flows” between compartments, as individuals transition from S to I to R.
Simulating the epidemic dynamics
Solving a system of ODEs for an SIR model allows us to approximate the total number of susceptible, infected, or recovered individuals in a population over time. This gives us a summary of epidemic dynamics at a larger scale.
The consequences of individual interactions determine epidemic dynamics at the population level.
In the figure below shows epidemic model outputs for a closed population with identical starting conditions, simulated at different transmission and recovery rates.
We can see that as the β and γ parameters are varied, the model predicts different outcomes:
There are several scenarios with no epidemic takeoff, when transmission (β) is slow and recovery (γ) is fast.
As β increases from left to right, we see curves with steeper slopes, higher peaks, earlier turnover, and shorter epidemics. Faster transmission rates means that the infection spreads more rapidly, but it also means that you quickly run out of susceptibles.
As γ decreases from top to bottom, we again see curves with steeper slopes and higher peaks, but later turnover and longer epidemics. This makes intuitive sense – if individuals stay infectious for a longer period, it takes longer for the epidemic to die out.
While these simulations help us understand the how each scenario was shaped by the parameters, we want to be able to anticipate trajectories without simulating the model outputs each time.
How can we anticipate trajectories without resorting to extensive numerical integration?
Calculating the invasion threshold for a disease can help us answer a key question:
Under what conditions will an infectious disease invade a system?
3.2 Mathematical derivation of R0 formula
Let’s review our most basic SIR compartmental model framework:
Individuals are categorized
into three compartments based on their infection status: Susceptible
(S), Infected (I), and Recovered (R).
Susceptible individuals have no immunity to the disease and will become infected upon successful transmission from an Infected individual. Infected individuals are those who have the pathogen and are capable of transmitting it to others (i.e., the model assumes that infected individuals are also infectious). Recovered individuals are those who have already been infected, and are now immune to reinfection.
The parameter \(\beta\) is the instantaneous transmission rate.
The parameter \(\gamma\) is the instantaneous recovery rate.
By definition, \(S + I + R = N\)
R0 can be calculated from the ODEs, without using numerical integration.
β for frequency-dependent transmission vs. density-dependent transmission
Density-dependent (Mass action): Number of contacts scales with the population density. Good for small population size, and directly transmitted diseases.
Transmission rate = \(S \times c(N) \times \frac{I}{N} \times \nu\)
Frequency-dependent: Contact rate is independent of population density. Good for large populations, vector borne diseases, or STIs.
Transmission rate = \(S \times c \times \frac{I}{N} \times \nu\)
3.2.1 R0 for density-dependent transmission
If we assume density-dependent transmission the transmission term is \(\beta S I\). Thus, the rates of change in state variables are described by these ODEs:
\(\frac{d S}{d t} = - \beta S I\)
\(\frac{d I}{d t} = \beta S I - \gamma I\)
\(\frac{d R}{d t} = \gamma I\)
For the disease to invade a population, we require \(\frac{dI}{dt} > 0\).
\[ \beta S I - \gamma I > 0 \]
We can rearrange that to get:
\[ \beta S I > \gamma I \]
This makes sense, because if you think back to the bathtub “flow” model, βSI is the inflow rate, and γI is the outflow rate. When the inflow rate is greater than the outflow rate, that’s when the epidemic is growing (the number of infecteds in increasing).
Remember that R0 is the expected number of cases generated by one infectious case in a population where all individuals are susceptible to infection. The definition assumes that no other individuals are infected or immune. Therefore, the initial conditions for calculating R0 are:
\(S(0) = N - 1 \approx N\)
\(I(0) = 1\)
\(R(0) = 0\)
To find \(R_0\), assume \(S = N\) and \(I = 1\), leading to the equation:
\[ \beta N > \gamma \]
Divide by \(\gamma\) to give:
\[ \frac{\beta N}{\gamma} > 1 \]
Thus, the condition for disease invasion is:
\[ \frac{\beta N}{\gamma} > 1 \]
This is known as the invasion threshold.
Therefore, R0 for the SIR model above with density-dependent transmission is calculated by this equation:
\[ R_0 = \frac{\beta N}{\gamma} \]
Practice question 2
A. Disease measles modeled by density-dependent transmission. True or false?
B. In a population of 500 individuals with measles, knowing that measles has an infectious period of about 14 days and a \(\beta\) of 0.002. Calculate \(R_0\).
3.2.2 R0 for frequency-dependent transmission
The rates of change in state variables are described by these differential equations:
- \(\frac{d S}{d t} = - \frac{\beta S I}{N}\)
- \(\frac{d I}{d t} = \frac{\beta S I}{N} - \gamma I\)
- \(\frac{d R}{d t} = \gamma I\)
For the disease to invade a population, we require \(\frac{dI}{dt} > 0\).
\[ \frac{\beta S I}{N} - \gamma I > 0 \]
We can rearrange that to get:
\[ \frac{\beta S I}{N} > \gamma I \]
Once again we can assume that \(S = N\) and \(I = 1\), leading to the condition:
\[ \beta > \gamma \]
Divide by \(\gamma\) to give:
\[ \frac{\beta}{\gamma} > 1 \]
This condition for disease invasion.
Therefore, R0 for the SIR model above with frequency-dependent transmission is calculated by this equation:
\[ R_0 = \frac{\beta}{\gamma} \]
Practice question 3
A. An STI would be modeled by density-dependent transmission. True or false?
B. In a population of 1200 individuals with HIV, with an average infectious period of 10 years and a \(\beta\) of 0.4. Calculate \(R_0\).
4 Implications of R0
For the SIR model, the ratio β/γ gives number of secondary cases will be infected before the index case recovers. Hence, R0 depends on both properties of the pathogen and properties of the population into which it is introduced.
R0 values help determine whether a disease will spread or decline in a community. if R0 is less than 1, the disease will die out in a population. On the other hand, if R0 is greater than 1, the disease will spread. The higher the R0, the faster the epidemic.
Model Predictions Deviate from Reality
So far in this course we have only been using deterministic models, to keep things simple. However, the assumptions of this theoretical model are virtually never met in real life. Reality is much more complex, and chance/randomness also plays a role in determining epidemic trajectories – the epidemic might go extinct when >1; and it’s also possible to have at least a small outbreak when <1).
Randomness can be accounted for with stochastic SIR models. The predicted trajectory of individual stochastic runs (each simulation) can vary widely, indicating a large number of different possible epidemic paths. However, the average of all the stochastic simulations will converge towards the deterministic prediction.
This is just an average, <1 means that if the situation is repeated many times, most outbreaks would go extinct.
4.1 Caveats/limitations
Not a universal property of a pathogen. It depends on the contact rates and contact structure. For example, the R0 for HIV among MSM population in London is drastically different than the R0 for HIV among female sex workers (FSW) in Kenya.
Another caveat to consider is that R0 is only appropriate under specific starting conditions at the beginning of an epidemic. As the disease spreads, the number of susceptible individuals decreases, and R0 becomes an overestimate (because the whole populations is no longer in the S category).
When the percentage of susceptibles decreases, there is a “wastage” of contact. Even though the average number of contacts per individual remains the same, the likelihood of contact between S and I individuals is decreasing.
As the epidemic grows, the effective reproduction number (Re), becomes a more accurate measure of disease transmission. We will look at Re in detail in the next lesson.
Super spreaders (TBA)
Conclusion
In this lesson, we explored the concept of invasion thresholds, the basic reproduction number (R0), and their role in understanding infectious disease dynamics. We learned to calculate R0 for a given infectious disease model. We also covered key implications of R0 and its limitations.
Answer Key
Practice Question 1
FALSE. R0 is an the average number of expected cases. For example, the following figure illustrates the beginning of an epidemic with an R0 of 2.
Practice Question 2
TRUE. Measles is an airborne pathogen which can be contracted by breathing the same air an infectious individual. The rate of contact is expected to increase with population density, therefore it is typically modeled assuming density-dependent transmission.
R0 is 14. For the simple density-dependent SIR model (with constant population size, no births or deaths), R0 is calculated by the equation:
\[ R_0 = \frac{\beta N}{\gamma} \]
Given that \(\beta\) = 0.002, and \(N\) = 500, we need to know \(\gamma\) in order to calculate \(R_0\). Recall that \(1/ \gamma\) is the mean value of the infectious period. There for, \(\gamma = 1/14 = 0.07142857\) and
\[ R_0 = \frac{0.002*500}{0.07142857} = 14 \]
Practice Question 3
- FALSE. The rate of sexual contact is not expected to increase with population density, therefore sexually-transmitted diseases are typically modeled assuming frequency-dependent transmission.
3B) CR0 is 4. For the simple frequency-dependent SIR model (with constant population size, no births or deaths), R0 is calculated by the equation:
\[ R_0 = \frac{\beta}{\gamma} \]
Given that \(\beta\) = 0.4, we need to know \(\gamma\) in order to calculate \(R_0\). Recall that \(1/ \gamma\) is the mean value of the infectious period. There for, \(\gamma = 1/10 = 0.1\) and
\[ R_0 = \frac{0.4}{0.1} = 4 \]
Contributors
The following team members contributed to this lesson:
JOY VAZ
Loves doing science and teaching science
SABINA RODRIGUEZ VELÁSQUEZ
Infectiously enthusiastic about microbes and Global Health
